Conventionally, methods such as the followings are known as production method for a star polymer having plural arms.
(a) A method wherein an AB-form block copolymer and an ABA-form block copolymer that present different properties such as regarding amphipathicity are subjected to micellation in a solvent, which are then directly cross-linked or their internal cores are cross-linked by some means (see for example, Patent Reference 1).(b) A method wherein arms are formed from the core compound by a polymer polymerization process such as a living polymerization process (see for example, Non-patent Reference 1).(c) A method wherein a dendrimer having a plurality of multiply branched chains is used (see for example, Patent Reference 2).
However, the method (a) in which block copolymers are used requires the micelle formation at the critical micelle concentration (CMC). Moreover, it is sometimes hard to form the micelle per se depending on the constitutional ratio of the polymers, and it is sometimes hard to establish an internal cross-linkage even when the micelle has been formed. As to the method (b) in which arms are formed from the core compound by a polymerization process, advanced polymerization techniques and a polymerization facility are required. As to the method (c) in which a dendrimer is used, it also requires advanced synthesis technique similarly as for the method (b) in spite that the dendrimer for use is a compound superior for its multiply branched property.
On the other hand, it is reported that a star-structured polymer which is obtained by the method comprising: subjecting methylmethacrylate, isobutylmethacrylate and t-butylmethacrylate to a polymerization by using diphenylhexyllithium as a polymerization initiator, wherein the diphenylhexyllithium is obtained by reacting 1,1-diphenylethylene and sec-butyllithium; and allowing the resultant to react with dicumylalcohol dimethacrylate or 2,5-dimethyl-2,5-hexandiordimethacrylate to obtain a star-structured polymer (see Non-patent Reference 2). However, star polymers obtained by conventional production methods including the above-mentioned method have a molecular weight distribution of 1.5 or more and it was difficult to form star polymers having uniform arms.
As a method for producing a hetero-armed star polymer, the method for forming a hetero-armed star copolymer is proposed which comprises the following steps: providing a first polymer having a first radically transferable atom or group; adding a coupling compound containing one or more α,α-disubstituted olefin group to the first polymer in the presence of a transition metal complex capable of undergoing a redox reaction with the first radically transferable atom or group; resulting in the addition of the coupling compound containing the α,α-disubstituted olefin group at the site of the first radically transferable atom or group and an elimination reaction comprising the radically transferable atom or group to form a reactive double bond; and allowing a second polymer having a second radically transferable atom or group in the presence of the transition metal complex to add to the reactive double bond. In this method, the first polymer and the second polymer are different from each other for at least either the molecular weight or the constitutional ratio (see Patent Reference 3).
However, the above method for producing a hetero-armed star polymer has drawbacks in that it sometimes causes a low yield and a wide molecular weight distribution, and that it is also difficult to obtain a polymer with the molecular weight as designed.    Patent Reference 1: Japanese Laid-Open Patent Application No. 10-195152    Patent Reference 2: Japanese Laid-Open Patent Application No. 6-219966    Patent Reference 3: Published Japanese translation of PCT International Publication No. 2002-540234    Non-patent Reference 1: Macromol. Chem., 189, 2885-2889 (1988)    Non-patent Reference 2: J. Polymer Science, Part A, 2003, 3083